Optimal. Leaf size=47 \[ \frac{1}{4} x^4 \sqrt{a+\frac{b}{x^4}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )}{4 \sqrt{a}} \]
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Rubi [A] time = 0.0256478, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 47, 63, 208} \[ \frac{1}{4} x^4 \sqrt{a+\frac{b}{x^4}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )}{4 \sqrt{a}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \sqrt{a+\frac{b}{x^4}} x^3 \, dx &=-\left (\frac{1}{4} \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^2} \, dx,x,\frac{1}{x^4}\right )\right )\\ &=\frac{1}{4} \sqrt{a+\frac{b}{x^4}} x^4-\frac{1}{8} b \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x^4}\right )\\ &=\frac{1}{4} \sqrt{a+\frac{b}{x^4}} x^4-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x^4}}\right )\\ &=\frac{1}{4} \sqrt{a+\frac{b}{x^4}} x^4+\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )}{4 \sqrt{a}}\\ \end{align*}
Mathematica [A] time = 0.1051, size = 62, normalized size = 1.32 \[ \frac{1}{4} x^2 \sqrt{a+\frac{b}{x^4}} \left (\frac{\sqrt{b} \sinh ^{-1}\left (\frac{\sqrt{a} x^2}{\sqrt{b}}\right )}{\sqrt{a} \sqrt{\frac{a x^4}{b}+1}}+x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 68, normalized size = 1.5 \begin{align*}{\frac{{x}^{2}}{4}\sqrt{{\frac{a{x}^{4}+b}{{x}^{4}}}} \left ({x}^{2}\sqrt{a{x}^{4}+b}\sqrt{a}+b\ln \left ({x}^{2}\sqrt{a}+\sqrt{a{x}^{4}+b} \right ) \right ){\frac{1}{\sqrt{a{x}^{4}+b}}}{\frac{1}{\sqrt{a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.88002, size = 292, normalized size = 6.21 \begin{align*} \left [\frac{2 \, a x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} + \sqrt{a} b \log \left (-2 \, a x^{4} - 2 \, \sqrt{a} x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} - b\right )}{8 \, a}, \frac{a x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} - \sqrt{-a} b \arctan \left (\frac{\sqrt{-a} x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{a x^{4} + b}\right )}{4 \, a}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.4214, size = 44, normalized size = 0.94 \begin{align*} \frac{\sqrt{b} x^{2} \sqrt{\frac{a x^{4}}{b} + 1}}{4} + \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a} x^{2}}{\sqrt{b}} \right )}}{4 \sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11436, size = 55, normalized size = 1.17 \begin{align*} \frac{1}{4} \, \sqrt{a x^{4} + b} x^{2} - \frac{b \log \left ({\left | -\sqrt{a} x^{2} + \sqrt{a x^{4} + b} \right |}\right )}{4 \, \sqrt{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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